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Abstract

The well-known greedy triangulation GT #S# of a fi#nite point set S is obtained by inserting compatible edges in increasing length order, where an edge is compatible if it does not cross previously inserted ones. Exploiting the concept of so-called light edges, we introduce a new way of defining GT #S#. The new definition does not rely on the length ordering of the edges. It provides a decomposition of GT #S#into levels, and the number of levels allows us to bound the total edge length of GT #S#. In particular, we show jGT #S#j#3#2 k+1 jMWT#S#j, where k is the number of levels and MWT#S# is the minimum-weight triangulation of S. This constitutes the first non-trivial upper bound on jGT #S#j for general points sets S.

Citations

...vex position [LL2], its worst-case length behaviour is fairly bad. The GT can be a factor of \Omega\Gamma p n) longer than the MWT; see [L]. Only very recently, a matching upper bound has been proved =-=[LK]-=-. In this note, we prove an upper bound of the form jGT (S)jsc k \Delta jMWT (S)j, where c k is a constant depending on the shape of S but not on its size. In Section 2, we introduce a decomposition o...

...uniformly distributed point sets [LL1] and for point sets in convex position [LL2], its worst-case length behaviour is fairly bad. The GT can be a factor of \Omega\Gamma p n) longer than the MWT; see =-=[L]-=-. Only very recently, a matching upper bound has been proved [LK]. In this note, we prove an upper bound of the form jGT (S)jsc k \Delta jMWT (S)j, where c k is a constant depending on the shape of S ...

...ength order, where an edge is compatible if it does not cross previously inserted ones. Various algorithms for computing the GT are known, and the GT has been used in several applications. See, e.g., =-=[DDMW]-=- for a short history. One use of the greedy triangulation is a length approximation to the minimum-weight triangulation (MWT). For a given point set S, the MWT minimizes the total edge length for all ...

... (p). 2 g q l r h p e Figure 3: Expansion of a star. Finally, in order to bound the weight of GT (S) by means of the weights of the points in S after step k, we utilize the following result proved in =-=[AAR]-=-. Lemma 6 Let T be an arbitrary triangulation of S. Then the edges of T can be oriented such that each point p 2 S has an in-degree of at most 3. The assertion below is now easy to prove. Lemma 7 jGT ...

...WT. Section 4 studies the number of levels in a GT and offers a short discussion of the presented topic. As we have learned recently, level decompositions of GTs have been considered independently in =-=[J]-=-, for the sake of an efficient parallel computation of greedy triangulations. An improved length bound of the form jGT (S)jsO(k) \Delta jMWT (S)j appeared to be implicit in the independent work [LK]. ...

...tions to the MWT are of importance. Although the GT tends to be short in practical applications, and is provably short for uniformly distributed point sets [LL1] and for point sets in convex position =-=[LL2]-=-, its worst-case length behaviour is fairly bad. The GT can be a factor of \Omega\Gamma p n) longer than the MWT; see [L]. Only very recently, a matching upper bound has been proved [LK]. In this note...